tokfandomcom-20200215-history
Lie derivative
In , the Lie derivative , named after by , evaluates the change of a (including scalar function, and ), along the defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any . Functions, tensor fields and forms can be differentiated with respect to a vector field. If T'' is a tensor field and ''X is a vector field, then the Lie derivative of T'' with respect to ''X is denoted \mathcal{L}_X(T) . The T \mapsto \mathcal{L}_X(T) is a of the algebra of of the underlying manifold. The Lie derivative commutes with and the on . Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or . Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function. The Lie derivative of a vector field Y'' with respect to another vector field ''X is known as the " " of X'' and ''Y, and is often denoted [X'',''Y] instead of \mathcal{L}_X(Y) . The space of vector fields forms a with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional of this Lie algebra, due to the identity : \mathcal{L}_{X,Y} T = \mathcal{L}_X \mathcal{L}_{Y} T - \mathcal{L}_Y \mathcal{L}_X T, valid for any vector fields X'' and ''Y and any tensor field T''. Considering vector fields as s of (i.e. one-dimensional of s) on ''M, the Lie derivative is the of the representation of the on tensor fields, analogous to Lie algebra representations as associated to in theory. Generalisations exist for fields, s with and vector-valued . Motivation A 'naïve' attempt to define the derivative of a with respect to a would be to take the of the tensor field and take the with respect to the vector field of each component. However, this definition is undesirable because it is not invariant under , e.g. the naive derivative expressed in polar or spherical coordinates differs from the naive derivative of the components in polar or spherical coordinates. On an abstract such a definition is meaningless and ill defined. In , there are three main coordinate independent notions of differentiation of tensor fields: Lie derivatives, derivatives with respect to , and the of completely anti symmetric (covariant) tensors or . The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a is well-defined even if it is not specified how to extend that tangent vector to a vector field. However a connection requires the choice of an additional geometric structure (e.g. a or just an abstract ) on the manifold. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X'' at a point ''p depends on the value of X'' in a neighborhood of ''p, not just at p'' itself. Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms (including functions). Definition The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors. The (Lie) derivative of a function The problem with generalizing the derivative of a function when we consider functions over manifolds is that the usual requires we define addition on the function's inputs but it is meaningless to add points on a manifold which is not a vector space. The critical issue, however, is to consider how the function changes relative to smooth displacements of the points. The Lie derivative of a scalar function can be thought of as a definition of the derivative where we are using the flows defined by vector fields to displace the points: :The Lie derivative of a function ''f with respect to a X'' at a point ''p of the manifold M'' is the value :: \mathcal{L}_X f (p) = \lim_{t\to 0} \frac{f(P(t;p)) - f(p)}{t} where P(t; p) is the point to which the defined by the vector field X maps the point p as follows: in a coordinate neighbourhood with coordinates x^1,..., x^n we can express the vector field (which we consider as a first order differential operator) as X = X^\mu(x)\partial_\mu = X^\mu \frac\partial{\partial x^\mu} . Locally, the flow is then defined by x^\mu(P(t; p)) = P^\mu(t, p^1,..., p^n) where p^\mu = x^\mu(p) and P^\mu(t;0) is the solution to the partial differential equation :: \partial_t P^\mu(t; p^1, ..., p^n) = X^\mu(P^1(t; p^1,..., p^n),..., P^n(t, p^1, ..., p^n)) with P^\mu(0; p^1,...,p^n) = p^\mu . We can then identify the Lie derivative of a function at ''p with the : :: (\mathcal{L}_X f)(p) = (X f)(p) = X^\mu \partial_\mu(f(p(x^1,...,x^n))) = \partial_t (f(P(t; p))) The Lie derivative of a vector field If X'' and ''Y are both vector fields, then the Lie derivative of Y'' with respect to ''X is also known as the of X'' and ''Y, and is sometimes denoted X,Y . There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above: * The Lie bracket of X'' and ''Y at p'' is given in local coordinates by the formula \mathcal{L}_X Y (p) = X,Y(p) = \partial_X Y(p) - \partial_Y X(p), where \partial_X and \partial_Y denote the operations of taking the s with respect to ''X and Y'', respectively. Here we are treating a vector in ''n-dimensional space as an n''- , so that its directional derivative is simply the tuple consisting of the directional derivatives of its coordinates. Note that although the final expression \partial_X Y(p) - \partial_Y X(p) appearing in this definition does not depend on the choice of local coordinates, the individual terms \partial_X Y(p) and \partial_Y X(p) do depend on the choice of coordinates. * If ''X and Y'' are vector fields on a manifold ''M according to the second definition, then the operator \mathcal{L}_X Y = X,Y defined by the formula :: X,Y: C^\infty(M) \rightarrow C^\infty(M) :: X,Y(f) = X(Y(f)) - Y(X(f)) is a derivation of order zero of the algebra of smooth functions of M'', i.e. this operator is a vector field according to the second definition. The Lie derivative of a tensor field More generally, if we have a ''T of (q,r) and a differentiable Y'' (i.e. a differentiable section of the ''TM), then we can define the Lie derivative of T'' along ''Y. Let, for some open interval I'' around 0, be the one-parameter semigroup of local diffeomorphisms of ''M induced by the of Y'' and denote . For each sufficiently small t, ''φ't'' is a diffeomorphism from a in M'' to another neighborhood in ''M, and φ''0 is the identity diffeomorphism. The Lie derivative of ''T is defined at a point p'' by : (\mathcal{L}_Y T)_p=\left.\frac{d}{dt}\right|_{t=0}\left((\varphi_{-t})_*T_{\varphi_{t}(p)}\right). where (\varphi_t)_* is the along the diffeomorphism and (\varphi_t)^* is the along the diffeomorphism. Intuitively, if you have a tensor field T and a vector field ''Y, then \mathcal{L}_{Y} T is the infinitesimal change you would see when you flow T using the vector field −''Y'', which is the same thing as the infinitesimal change you would see in T if you yourself flowed along the vector field Y''. We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms: :'Axiom 1.' The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula :: \mathcal{L}_Yf=Y(f) :'Axiom 2.' The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields ''S and T'', we have :: \mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\mathcal{L}_YT). :'Axiom 3.' The Lie derivative obeys the Leibniz rule with respect to : :: \mathcal{L}_X (T(Y_1, \ldots, Y_n)) = (\mathcal{L}_X T)(Y_1,\ldots, Y_n) + T((\mathcal{L}_X Y_1), \ldots, Y_n) + \cdots + T(Y_1, \ldots, (\mathcal{L}_X Y_n)) :'Axiom 4.' The Lie derivative commutes with exterior derivative on functions: :: d = 0 If these axioms hold, then applying the Lie derivative \mathcal{L}_X to the relation df(Y) = Y(f) shows that :: \mathcal{L}_X Y (f) = X(Y(f)) - Y(X(f)), which is one of the standard definitions for the . The Lie derivative of a differential form is the of the with the exterior derivative. So if α is a differential form, :: \mathcal{L}_Y\alpha=i_Yd\alpha+di_Y\alpha. This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions. Explicitly, let ''T be a tensor field of type . Consider T'' to be a differentiable of ''α''1, ''α''2, ..., ''αq'' of the cotangent bundle ''T∗''M'' and of sections X''1, ''X''2, ..., ''X''p of the ''TM, written T''(α''1, α''2, ..., ''X''1, ''X''2, ...) into '''R'. Define the Lie derivative of T'' along ''Y by the formula : (\mathcal{L}_Y T)(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) =Y(T(\alpha_1,\alpha_2,\ldots,X_1,X_2,\ldots)) :: - T(\mathcal{L}_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) - T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots :: - T(\alpha_1, \alpha_2, \ldots, \mathcal{L}_YX_1, X_2, \ldots) - T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_2, \ldots) - \ldots The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the for differentiation. Note also that the Lie derivative commutes with the contraction. The Lie derivative of a differential form A particularly important class of tensor fields is the class of . The restriction of the Lie derivative to the space of differential forms is closely related to the . Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an , after which the relationships falls out as an identity known as Cartan's formula. Note that Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms. Let M'' be a manifold and ''X a vector field on M''. Let \omega \in \Lambda^{k+1}(M) be a - , i.e. for each p \in M , \omega(p) is an from (T_p M)^{k + 1} to the real numbers. The of X'' and ''ω is the k''-form i_X\omega defined as : (i_X\omega) (X_1, \ldots, X_k) = \omega (X,X_1, \ldots, X_k)\, The differential form i_X\omega is also called the '''contraction' of ω'' with ''X. Note that : i_X:\Lambda^{k+1}(M) \rightarrow \Lambda^k(M) and that i_X is a \wedge - . That is, i_X is R'''-linear, and : i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta) for \omega \in \Lambda^k(M) and η another differential form. Also, for a function f \in \Lambda^0(M) , that is, a real- or complex-valued function on M'', one has : i_{fX} \omega = f\,i_X\omega where f X denotes the product of ''f and X''. The relationship between s and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function ''f with respect to a vector field X'' is the same as the directional derivative ''X(f''), it is also the same as the of the exterior derivative of ''f with X'': : \mathcal{L}_Xf = i_X \, df For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in ''X: : \mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega). This identity is known variously as '''Cartan formula, Cartan homotopy formula or Cartan's magic formula. See for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that : d\mathcal{L}_X\omega = \mathcal{L}_X(d\omega). The Lie derivative also satisfies the relation : \mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega . Coordinate expressions In local notation, for a type tensor field T , the Lie derivative along X is : \begin{align} (\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = & X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) \\ & - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} \\ & + (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} \end{align} here, the notation \partial_a = \frac{\partial}{\partial x^a} means taking the partial derivative with respect to the coordinate x^a . Alternatively, if we are using a (e.g., the ), then the partial derivative \partial_a can be replaced with the which means replacing \partial_a X^b with (by abuse of notation) \nabla_a X^b = X^b_{;a} := (\nabla X)_a^{\ b} = \partial_a X^b + \Gamma^b_{ac}X^c where the \Gamma^a_{bc} = \Gamma^a_{cb} are the . The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor (\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}\partial_{a_1}\otimes\cdots\otimes\partial_{a_r}\otimes dx^{b_1}\otimes\cdots\otimes dx^{b_s} which is independent of any coordinate system and of the same type as T . The definition can be extended further to tensor densities. If T'' is a tensor density of some real number valued weight ''w (e.g. the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight. : (\mathcal {L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} = X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} + :: + (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} + w (\partial_{c} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s}} Notice the new term at the end of the expression. For a \Gamma =( \Gamma^{a}_{bc} ) , the Lie derivative along X is : (\mathcal{L}_X \Gamma)^{a}_{bc} = X^d\partial_d \Gamma^{a}_{bc} + \partial_b\partial_c X^a - \Gamma^{d}_{bc}\partial_d X^a + \Gamma^{a}_{dc}\partial_b X^d + \Gamma^{a}_{bd}\partial_c X^d Examples For clarity we now show the following examples in local notation. For a \phi(x^c)\in\mathcal{F}(M) we have: : (\mathcal {L}_X \phi)=X(\phi) = X^a \partial_a \phi . Hence for the scalar field \phi(x,y) = x^2 - \sin(y) and the vector field X = \sin(x)\partial_y - y^2\partial_x the corresponding Lie derivative becomes \begin{alignat}{4} \mathcal{L}_X\phi &= (\sin(x)\partial_y - y^2\partial_x)(x^2 - \sin(y))\\ & = -\sin(x)\cos(y) - 2xy^2 \end{alignat} For an example of higher rank differential form, consider the 2-form \omega = (x^2 + y^2)dx\wedge dz and the vector field X from the previous example. Then, \begin{align} \mathcal{L}_X\omega & = d(i_{\sin(x)\partial_y - y^2\partial_x}((x^2 + y^2)dx\wedge dz)) + i_{\sin(x)\partial_y - y^2\partial_x}(d((x^2 + y^2)dx\wedge dz)) \\ & = d(-y^2(x^2 + y^2) dz) + i_{\sin(x)\partial_y - y^2\partial_x}(2ydy\wedge dx\wedge dz) \\ & = \left(- 2xy^2 dx + (-2yx^2 - 4y^3) dy\right) \wedge dz + (2y\sin(x)dx \wedge dz + 2y^3dy \wedge dz)\\ & = \left(-2xy^2 + 2y\sin(x)\right)dx\wedge dz + (-2yx^2 - 2y^3)dy\wedge dz \end{align} Some more abstract examples. Note that : \mathcal{L}_X (dx^b) = d i_X (dx^b) = d X^b = \partial_a X^b dx^a . Hence for a , i.e., a , A=A_a(x^b)dx^a we have: : \mathcal{L}_X A = X (A_a) dx^a + A_b \mathcal{L}_X (dx^b) = (X^b \partial_b A_a + A_b\partial_a (X^b))dx^a Note that the coefficient of the last expression is the local coordinate expression of the Lie derivative. For a covariant rank 2 tensor field T=T_{ab}(x^c)dx^a\otimes dx^b we have: \begin{align} (\mathcal {L}_X T) &= (\mathcal {L}_X T)_{ab} dx^a\otimes dx^b\\ &= X(T_{ab})dx^a\otimes dx^b + T_{cb} \mathcal{L}_X (dx^c) \otimes dx^b + T_{ac} dx^a \otimes \mathcal{L}_X (dx^c)\\ &= (X^c \partial_c T_{ab}+T_{cb}\partial_a X^c+T_{ac}\partial_b X^c)dx^a\otimes dx^b\\ \end{align} If T = g is the symmetric metric tensor, it is parallel with respect to the Levi Civita connection (aka covariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving : (\mathcal {L}_X g) = (X^c g_{ab; c}+g_{cb}X^c_{;a}+g_{ac}X^c_{; b})dx^a\otimes dx^b = (X_{b;a} + X_{a;b}) dx^a\otimes dx^b Properties The Lie derivative has a number of properties. Let \mathcal{F}(M) be the of functions defined on the M''. Then : \mathcal{L}_X : \mathcal{F}(M) \rightarrow \mathcal{F}(M) is a on the algebra \mathcal{F}(M) . That is, \mathcal{L}_X is '''R'-linear and : \mathcal{L}_X(fg)=(\mathcal{L}_Xf) g + f\mathcal{L}_Xg. Similarly, it is a derivation on \mathcal{F}(M) \times \mathcal{X}(M) where \mathcal{X}(M) is the set of vector fields on M'' (cf. Theorem 6 from the article: Nichita, F.F. Unification Theories: New Results and Examples. Axioms 2019, 8, 60): : \mathcal{L}_X(fY)=(\mathcal{L}_Xf) Y + f\mathcal{L}_X Y which may also be written in the equivalent notation : \mathcal{L}_X(f\otimes Y)= (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y where the symbol \otimes is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold. Additional properties are consistent with that of the . Thus, for example, considered as a derivation on a vector field, : \mathcal{L}_X Y,Z = Y,Z + Z one finds the above to be just the . Thus, one has the important result that the space of vector fields over ''M, equipped with the Lie bracket, forms a . The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M'', and let ''X and Y'' be two vector fields. Then * \mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha) \wedge\beta + \alpha\wedge (\mathcal{L}_X\beta) * \mathcal{L}_X,\mathcal{L}_Y\alpha:= \mathcal{L}_X\mathcal{L}_Y\alpha-\mathcal{L}_Y\mathcal{L}_X\alpha=\mathcal{L}_{X,Y}\alpha * \mathcal{L}_X,i_Y\alpha=i_X,\mathcal{L}_Y\alpha=i_{X,Y}\alpha, where ''i denotes interior product defined above and it's clear whether ·,· denotes the or the . Generalizations Various generalizations of the Lie derivative play an important role in differential geometry. Covariant Lie derivative If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle. Now, if we're given a vector field Y'' over ''M (but not the principal bundle) but we also have a over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches Y'' and its vertical component agrees with the connection. This is the covariant Lie derivative. See for more details. Nijenhuis–Lie derivative Another generalization, due to , allows one to define the Lie derivative of a differential form along any section of the bundle Ω''k(M'', T''M) of differential forms with values in the tangent bundle. If K'' ∈ Ω''k(M'', T''M) and α is a differential p''-form, then it is possible to define the interior product ''iK''α of ''K and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative: : \mathcal{L}_K\alpha=d,i_K\alpha = di_K\alpha-(-1)^{k-1}i_K \, d\alpha. History In 1931, introduced a new differential operator, later called by that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms. The Lie derivatives of general geometric objects (i.e., sections of s) were studied by , Y. Tashiro and . For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, —and before him (in 1921) —introduced what he called a ‘local variation’ \delta^{\ast}A of a geometric object A\, induced by an infinitesimal transformation of coordinates generated by a vector field X\, . One can easily prove that his \delta^{\ast}A is - \mathcal{L}_X(A)\, . References Category:Advanced mathematics